denote the usual operator norm. For
positive semidefinite matrices, this corresponds to the maximum
Question 1. What values can
take if are positive semidefinite?
Note that by convexity and submultiplicativity, we have
Clearly, the latter bound is tight (e.g., take ).
In this post, we will consider the first bound more carefully and show:
Proposition 1. Suppose . Let
For example, the above proposition implies that for any
, we have
Proof. By homogeneity, it suffices to consider the case where
Without loss of generality, each is a rank-one matrix. Indeed,
suppose . Let
correspond to a minimum eigenvalue of
Note that .
Then The last inequality
follows from the assumption that .
We may thus write each for some
. Equivalently, minimizes
By negating if necessary, we may assume that
and . By optimality and our
assumptions on the signs, we have that for each ,
We deduce that
that without loss of generality
. In particular, we may
for some and . Then,
This expression is minimized at where